Question: For which positive integer values of $k$ does $kx^2+20x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order.
Solution: By considering the expression $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for the solutions of $ax^2+bx+c=0$, we find that the solutions are rational if and only if the discriminant $b^2-4ac$ has a rational square root. Therefore, the solutions of $kx^2+20x+k=0$ are rational if and only if $400-4(k)(k)$ is a perfect square. (Recall that if $n$ is an integer which is not a perfect square, then $\sqrt{n}$ is irrational).  By writing the discriminant as $4(100-k^2)$, we see that we only need to check the integers $1\leq k\leq 10$.  Of these, $\boxed{6, 8\text{, and }10}$ work.